relation between regularity and Fourier coefficient (reference)

Question

let $f$ a real continuous 2-$\pi$ periodic function and $\sum\limits_{n=-\infty}^{\infty} C_n(f)e^{inx} $ the Fourier series of $f$

I want some books like a reference of this equivalence.

$f$ is $C^k$ if and only if $C_n(f)=O(1/(n^k)). $

Thanks in advance

Answer 1

Look up page 101, 102 of the book Fourier Analysis and Partial Differential Equations by Rafael Iorio/Valeria De Magalhaes Iorio.

The proof is essentially by integration by parts and induction and the fact that $|(f')^{\wedge}(k)|\leq\|f'\|_{L^{1}}$.